Complex genetic diseases exhibit greater or lesser degrees of familial aggregation but do not fit simple Mendelian patterns of inheritance. Presumably, genes and environment both contribute to the occurrence of these diseases. Since genes act in only a few, well-understood ways, it seems logical to identify genetic factors first, then use them to sort out environmental contributions. Gene linkage analysis and segregation analysis represent two powerful tools for identifying specific genes, detecting genetic heterogeneity, and predicting recurrence risk. I propose continuing to attack several methodological and mathematical problems relating to the genetic epidemiology of common and/or complex diseases. As each problem is resolved, and new methodologies are developed, I will apply the resulting solution to data on four diseases: insulin dependent diabetes (IDDM), multiple sclerosis, ankylosing spondylitis, and coeliac disease. Better understanding of the genetics of these diseases will lead eventually to better treatment, prevention, or even cure. Work will focus on three problems relating to gene linkage analysis -- disease-marker associations (HLA) and disease), two-locus linkage analysis, and variable age-of-onset -- and two problems relating to segregation analysis -- genetic heterogeneity and ascertainment corrections. Then I will turn to theoretical and practical issues (sample size, robustness, asymptotic behavior) associated with the use of likelihood methods in genetic epidemiology. Moreover, the project will not necessarily be restricted to the specific areas detailed above. As time permits, and as particular problems arise, I will devote effort to them. Two problems that could arise are gene mapping functions and multilocus gene mapping, for example.